Augene: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
→Tuning spectrum: some of the edo tunings outright don't support augene. Un-normalize ratios to the octave-reduced form. +note
Rework the interval table
Line 3: Line 3:
The first few augene [[mos scale]]s are [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]], … and the first edos that reasonably support augene are [[12edo]], [[15edo]], and especially [[27edo]].
The first few augene [[mos scale]]s are [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]], … and the first edos that reasonably support augene are [[12edo]], [[15edo]], and especially [[27edo]].


== 11-limit interval chain ==
== Interval chain ==
{| class="wikitable center-1 right-all"
In the following table, odd harmonics 1–11 are in '''bold'''.
|+Generator based off of 11-limit minimax
 
! Gen.
{| class="wikitable center-1 right-2 right-4 right-6"
! Period 0
! rowspan="2" | #
! Period 1
! colspan="2" | Period 0
! Period 2
! colspan="2" | Period 1
! colspan="2" | Period 2
|-
! Cents*
! Approx. Ratios
! Cents*
! Approx. Ratios
! Cents*
! Approx. Ratios
|-
|-
| 0
| 0
| 0.000
| 0.0
| 400.000
| 1/1
| 800.000
| 400.0
| '''5/4''', 14/11
| 800.0
| '''8/5''', 11/7
|-
|-
| 1
| 1
| 1113.148
| 1113.1
| 313.148
| 15/8, 40/21
| 713.148
| 313.1
| 6/5
| 713.1
| '''3/2'''
|-  
|-  
| 2
| 2
| 1026.296
| 1026.3
| 226.296
| 9/5, 20/11
| 626.296
| 226.3
| '''8/7''', '''9/8'''
| 626.3
| 10/7, '''16/11'''
|-
|-
| 3
| 3
| 939.444
| 939.4
| 139.444
| 12/7
| 539.444
| 139.4
| 12/11, 15/14
| 539.4
| 15/11
|-
|-
| 4
| 4
| 852.592
| 852.6
| 52.592
| 18/11
| 452.592
| 52.6
| 45/44
| 452.6
| 9/7
|-
|-
| 5
| 5
| 765.740
| 765.7
| 1165.740
| 54/35
| 365.740
| 1165.7
| 27/14, 96/49
| 365.7
| 27/22
|}
|}
<nowiki>*</nowiki> in 11-odd-limit minimax tuning


== Tuning spectrum ==
== Tuning spectrum ==

Revision as of 14:32, 13 May 2024

Augene is a regular temperament of the augmented family, which means that 128/125 is tempered out and three 5/4's make a 2/1. Augene is distinguished from its relative august by tempering out 64/63, which means that tunings in which the fifth is larger than 7\12 (700 cents) are optimal. Therefore the optimal ET sequence of augene goes 12, 15, 27, … in constrast to august's which goes 9, 12, 21, ….

The first few augene mos scales are 3L 3s, 3L 6s, 3L 9s, 12L 3s, … and the first edos that reasonably support augene are 12edo, 15edo, and especially 27edo.

Interval chain

In the following table, odd harmonics 1–11 are in bold.

# Period 0 Period 1 Period 2
Cents* Approx. Ratios Cents* Approx. Ratios Cents* Approx. Ratios
0 0.0 1/1 400.0 5/4, 14/11 800.0 8/5, 11/7
1 1113.1 15/8, 40/21 313.1 6/5 713.1 3/2
2 1026.3 9/5, 20/11 226.3 8/7, 9/8 626.3 10/7, 16/11
3 939.4 12/7 139.4 12/11, 15/14 539.4 15/11
4 852.6 18/11 52.6 45/44 452.6 9/7
5 765.7 54/35 1165.7 27/14, 96/49 365.7 27/22

* in 11-odd-limit minimax tuning

Tuning spectrum

Edo
Generator 		Eigenmonzo
(Unchanged-interval)* 		Fifth (¢) Comments
13/7 671.702
15/8 688.269
13/11 689.210
7\12 700.000
3/2 701.955
30\51 705.882
15/14 706.481
23\39 707.692
7/5 708.744
9/7 708.771
9/5 708.798 9-odd-limit minimax
39\66 709.091
7/6 711.043
16\27 711.111
15/11 712.317
57\96 712.500
11/9 713.148 11-odd-limit minimax
25\42 714.286
7/4 715.587 7-, 13- and 15-odd-limit minimax
5/3 715.641 5-odd-limit minimax
34\57 715.789
25\72 716.667
11/6 716.879
11/10 717.498
9\15 720.000
13/9 721.127
15/13 723.871
11/8 724.341
13/12 730.714
13/10 745.786
13/8 759.472

* besides the octave

Music

Igliashon Jones
Joel Grant Taylor
Retrieved from "https://en.xen.wiki/index.php?title=Augene&oldid=143068"